In this section we describe a general framework of \(L\) -functions over function fields that will be used as generating series for the arithmetic functions under consideration. For a general reference, see [1]}. We consider the ring of polynomials \(\mathbb {F}_q[T]\) , where \(q\)  is an odd prime power. Let \(\mathcal {M}\)  denote the set of monic polynomials in \(\mathbb {F}_q[T]\) , and \(\mathcal {M}_n\)  the subset of \(\mathcal {M}\)  of polynomials of degree \(n\) . Similarly, let \(\mathcal {P}\)  denote the set of monic irreducible polynomials over \(\mathbb {F}_q[T]\) , and \(\mathcal {P}_n\)  the corresponding subset of irreducible polynomials of degree \(n\) . The norm of a polynomial \(f\in \mathbb {F}_q[T]\)  is defined as \(|f|:=q^{\deg (f)}\) . The zeta function for \(\mathbb {F}_q[T]\)  is given by
\(\zeta _q(s)=\sum _{f\in \mathcal {M}} \frac{1}{|f|^s}=\prod _{P\in \mathcal {P}} \left(1-\frac{1}{|P|^s}\right)^{-1} = \frac{1}{1-q^{1-s}}.\) 
